Optimal. Leaf size=131 \[ \frac {c (c x)^{4/3} \sqrt [3]{a+b x^2}}{2 b}+\frac {a c^{7/3} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{b} (c x)^{2/3}}{c^{2/3} \sqrt [3]{a+b x^2}}}{\sqrt {3}}\right )}{\sqrt {3} b^{5/3}}+\frac {a c^{7/3} \log \left (\sqrt [3]{b} (c x)^{2/3}-c^{2/3} \sqrt [3]{a+b x^2}\right )}{2 b^{5/3}} \]
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Rubi [A]
time = 0.10, antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {327, 335, 281,
337} \begin {gather*} \frac {a c^{7/3} \text {ArcTan}\left (\frac {\frac {2 \sqrt [3]{b} (c x)^{2/3}}{c^{2/3} \sqrt [3]{a+b x^2}}+1}{\sqrt {3}}\right )}{\sqrt {3} b^{5/3}}+\frac {a c^{7/3} \log \left (\sqrt [3]{b} (c x)^{2/3}-c^{2/3} \sqrt [3]{a+b x^2}\right )}{2 b^{5/3}}+\frac {c (c x)^{4/3} \sqrt [3]{a+b x^2}}{2 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 281
Rule 327
Rule 335
Rule 337
Rubi steps
\begin {align*} \int \frac {(c x)^{7/3}}{\left (a+b x^2\right )^{2/3}} \, dx &=\frac {c (c x)^{4/3} \sqrt [3]{a+b x^2}}{2 b}-\frac {\left (2 a c^2\right ) \int \frac {\sqrt [3]{c x}}{\left (a+b x^2\right )^{2/3}} \, dx}{3 b}\\ &=\frac {c (c x)^{4/3} \sqrt [3]{a+b x^2}}{2 b}-\frac {(2 a c) \text {Subst}\left (\int \frac {x^3}{\left (a+\frac {b x^6}{c^2}\right )^{2/3}} \, dx,x,\sqrt [3]{c x}\right )}{b}\\ &=\frac {c (c x)^{4/3} \sqrt [3]{a+b x^2}}{2 b}-\frac {(a c) \text {Subst}\left (\int \frac {x}{\left (a+\frac {b x^3}{c^2}\right )^{2/3}} \, dx,x,(c x)^{2/3}\right )}{b}\\ &=\frac {c (c x)^{4/3} \sqrt [3]{a+b x^2}}{2 b}-\frac {(a c) \text {Subst}\left (\int \frac {x}{1-\frac {b x^3}{c^2}} \, dx,x,\frac {(c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )}{b}\\ &=\frac {c (c x)^{4/3} \sqrt [3]{a+b x^2}}{2 b}-\frac {\left (a c^{5/3}\right ) \text {Subst}\left (\int \frac {1}{1-\frac {\sqrt [3]{b} x}{c^{2/3}}} \, dx,x,\frac {(c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )}{3 b^{4/3}}+\frac {\left (a c^{5/3}\right ) \text {Subst}\left (\int \frac {1-\frac {\sqrt [3]{b} x}{c^{2/3}}}{1+\frac {\sqrt [3]{b} x}{c^{2/3}}+\frac {b^{2/3} x^2}{c^{4/3}}} \, dx,x,\frac {(c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )}{3 b^{4/3}}\\ &=\frac {c (c x)^{4/3} \sqrt [3]{a+b x^2}}{2 b}+\frac {a c^{7/3} \log \left (c^{2/3}-\frac {\sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )}{3 b^{5/3}}+\frac {\left (a c^{5/3}\right ) \text {Subst}\left (\int \frac {1}{1+\frac {\sqrt [3]{b} x}{c^{2/3}}+\frac {b^{2/3} x^2}{c^{4/3}}} \, dx,x,\frac {(c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )}{2 b^{4/3}}-\frac {\left (a c^{7/3}\right ) \text {Subst}\left (\int \frac {\frac {\sqrt [3]{b}}{c^{2/3}}+\frac {2 b^{2/3} x}{c^{4/3}}}{1+\frac {\sqrt [3]{b} x}{c^{2/3}}+\frac {b^{2/3} x^2}{c^{4/3}}} \, dx,x,\frac {(c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )}{6 b^{5/3}}\\ &=\frac {c (c x)^{4/3} \sqrt [3]{a+b x^2}}{2 b}+\frac {a c^{7/3} \log \left (c^{2/3}-\frac {\sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )}{3 b^{5/3}}-\frac {a c^{7/3} \log \left (c^{4/3}+\frac {b^{2/3} (c x)^{4/3}}{\left (a+b x^2\right )^{2/3}}+\frac {\sqrt [3]{b} c^{2/3} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )}{6 b^{5/3}}-\frac {\left (a c^{7/3}\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{b} (c x)^{2/3}}{c^{2/3} \sqrt [3]{a+b x^2}}\right )}{b^{5/3}}\\ &=\frac {c (c x)^{4/3} \sqrt [3]{a+b x^2}}{2 b}+\frac {a c^{7/3} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{b} (c x)^{2/3}}{c^{2/3} \sqrt [3]{a+b x^2}}}{\sqrt {3}}\right )}{\sqrt {3} b^{5/3}}+\frac {a c^{7/3} \log \left (c^{2/3}-\frac {\sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )}{3 b^{5/3}}-\frac {a c^{7/3} \log \left (c^{4/3}+\frac {b^{2/3} (c x)^{4/3}}{\left (a+b x^2\right )^{2/3}}+\frac {\sqrt [3]{b} c^{2/3} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )}{6 b^{5/3}}\\ \end {align*}
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Mathematica [A]
time = 1.24, size = 174, normalized size = 1.33 \begin {gather*} \frac {(c x)^{7/3} \left (3 b^{2/3} x^{4/3} \sqrt [3]{a+b x^2}+2 \sqrt {3} a \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{b} x^{2/3}}{\sqrt [3]{b} x^{2/3}+2 \sqrt [3]{a+b x^2}}\right )+2 a \log \left (-\sqrt [3]{b} x^{2/3}+\sqrt [3]{a+b x^2}\right )-a \log \left (b^{2/3} x^{4/3}+\sqrt [3]{b} x^{2/3} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}\right )\right )}{6 b^{5/3} x^{7/3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (c x \right )^{\frac {7}{3}}}{\left (b \,x^{2}+a \right )^{\frac {2}{3}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 18.73, size = 44, normalized size = 0.34 \begin {gather*} \frac {c^{\frac {7}{3}} x^{\frac {10}{3}} \Gamma \left (\frac {5}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {2}{3}, \frac {5}{3} \\ \frac {8}{3} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {2}{3}} \Gamma \left (\frac {8}{3}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (c\,x\right )}^{7/3}}{{\left (b\,x^2+a\right )}^{2/3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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